Couple of things.
1. My Dad had a melanoma removed from his back, the wound was the size of your fist - deep as well as tall and wide. It did heal over, but that was a LONG time ago and I have no idea what he did - different medicine back then too.
2. That big of an open wound will probably drain for a few days, maybe a week? I however am not a doctor, although I did play one with Suzie when I was a kid!
3. Get a second - or third - opinion, get your personal physician involved. The "no antibiotics" thing on an open wound bothers me a bit, that doesn't sound normal. However, now that the cutter has had his say, most other doctors probably won't interfere. But I would ask why no antibiotics and demand an reasonable answer.
4. Always always
always!!!!! take control of your medical procedures. To the doctors, you are another stat to be dealt with clinically and efficiently. NO one - and I mean no one - will be looking out for you but you (OK, and Melanie in this case). Ask questions, probe the answers, look online, get the facts, regardless of how minor or trivial the procedure looks. Be a pest to the doctors so they will answer you just to get rid of you.
It is a proven fact that people who approach medicine as I suggested in #4 do far better than those who put their lives in their doctor's hands.